I've never tried to run the game as a non-admin like you're doing here. Your best bet is probably to try to install the game in a directory (folder) that does not require admin privileges for write access, such as under "My Documents" or the desktop. I don't remember however whether the Lemmings Revolution installer lets you specify where to install the game to. I guess it does but I don't know for sure off top of my head. Also, I can't guarantee that it's sufficient since I don't know for sure what operation (it may actually be the registry and not files) the game is being denied access to under a non-admin account.
On Vista and Win7, I believe you can set compatibility settings that address this sort of issue--Windows under the covers will redirect the game's attempts to read/write to protected locations (file and registry) into some other virtualized locations that a non-admin account can access. But I don't think XP has that. Assuming the problem was due to the game trying to write to one of its files but the files are in a directory requiring admin privileges to access, installing the game in a different directory should hopefully help.